The phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of 4. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from those of the square-lattice Ising model. An exponential decay is observed in the density-matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.
The phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of 4. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from those of the square-lattice Ising model. An exponential decay is observed in the density-matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.
The critical behavior of the classical Ising model on a three-dimensional fractal lattice with Hausdorff dimension dH=ln32/ln4=2.5 is investigated using the higher-order tensor renormalization group (HOTRG) method. We determine the critical temperature Tc≈2.65231 and the critical exponents for magnetization β≈0.059 and field response δ≈35. Unlike a previously studied 2D fractal with dH≈1.792, the specific heat for this 3D fractal exhibits a divergent singularity at Tc. The results are compared with those for regular lattices and other fractal structures to elucidate the role of dimensionality in critical phenomena.
The Ising model on fractal lattices has attracted renewed interest, yet a quantitative understanding of how the critical temperature depends on the Hausdorff dimension (D_{H}) and connectivity (Q) remains limited. We study the ferromagnetic Ising model on the one-parameter "legs-extension" family of fractal lattices originally introduced by Genzor et al. and apply the higher-order tensor renormalization group (HOTRG) method, following the tensor-network construction developed for these geometries in earlier work. By tuning a single structural parameter, we access lattices with D_{H} decreasing from 1.792 to 1.260 and connectivity Q from 0.50 to 0.131, and obtain high-precision critical temperatures T_{c} and magnetization exponents β with carefully quantified uncertainties from systematic bond-dimension and fitting-window analyses, finding that β is strongly and almost exponentially suppressed as D_{H} is reduced. Along this one-dimensional trajectory in the (D_{H},Q) plane, T_{c} decreases monotonically with D_{H} and, when plotted against Q, exhibits a robust near-linear behavior within this lattice family over almost an order of magnitude in connectivity. A finite-temperature phase transition persists throughout the explored range down to our lowest Hausdorff dimension, D_{H}≈1.260, implying that the lower critical dimension of this family, if it exists, lies below this value. Comparison with zero-order and first-order mean-field theory shows that HOTRG systematically incorporates fluctuation effects beyond mean-field approximations. The tabulated T_{c} and β values thus provide a stringent benchmark for theories of criticality on fractal lattices and a quantitative reference for future numerical and analytical studies on other complex geometries.
The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ, and δ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.
The transverse-field Ising model on the Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski fractal, which is characterized by the fractal dimension ${log}_{2}^{\phantom{\rule{3.33333pt}{0ex}}}3\ensuremath{\approx}1.585$, is studied by a tensor-network method known as the higher-order tensor renormalization group. We analyze the ground-state energy and the spontaneous magnetization in the thermodynamic limit. The system exhibits the second-order phase transition at the critical transverse field hc=1.865. The critical exponents $\ensuremath{\beta}\ensuremath{\approx}0.198$ and $\ensuremath{\delta}\ensuremath{\approx}8.7$ are obtained. Complementary to the tensor-network method, we make use of the real-space renormalization group and improved mean-field approximations for comparison.
Adopting concurrence as entanglement measure, we study entanglement and quantum phase transition of the Ising spin systems on the triangular lattice and Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski fractal lattices by using the quantum renormalization-group method. It is found that the ground-state entanglement between two spins (or spin blocks) depends on the following factors: the size of system, the magnetic field, the exchange coupling, and the structure of lattice. As the size of the system becomes large, (a) the range of the magnetic field, in which the entanglement exists, contracts gradually and focuses on the critical point; and (b) the first derivative of entanglement shows singular behavior, and its maximum or minimum is approaching to the critical point gradually. The scaling behaviors of entanglement on the different lattices are similar but the scaling relations are diverse. For the triangular lattice, the space dimensionality determines the scaling relationship between the critical exponent of the entanglement and the critical exponent correlation length. However, for fractal lattices, it is the fractal dimensionality but not the space one to determine this relationship.
Abstract The critical behavior of spin systems is fundamentally governed by dimensionality and connectivity. Moving beyond translationally invariant lattices, we explore a new paradigm where fractality itself becomes a tunable parameter, engineering magnetic order and critical behavior. By implementing the classical Ising model on a three-dimensional fractal lattice---a static realization of a spin cluster with Hausdorff dimension dH=2.5 and boundary dimension d=2---we demonstrate how fractal geometry dictates unique critical phenomena. Using the higher-order tensor renormalization group (HOTRG) method, we identify a finite-temperature phase transition at Tc≈2.65231 with exotic critical exponents (α≈0.09, α′≈0.06, β≈0.059, δ≈35) and a diverging specific heat---a hallmark absent in lower-dimensional fractals. This work establishes that fractal geometry serves as a powerful and untapped degree of freedom for spintronics. It provides a blueprint for designing materials with programmable magnetic phase transitions, paving the way for next-generation, geometry-driven devices in magnonics and neuromorphic computing.